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Alternating-Current Circuits

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Topics

Alternating Voltages and Currents

Capacitors in AC Circuits

RCCircuits

Inductors in AC Circuits

RLCCircuits

Resonance in Electrical Circuits

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Wall sockets provide current and voltage that

vary sinusoidally with time.

ere is a simple ac circuit!

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The voltage as a "unction o" time is!

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#ince this circuit has only a resistor$ the

current is given %y!

ere$ the current andvoltage have peaks

at the same time &

they are in phase.

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In order to visuali'e the phase relationships

%etween the current and voltage in ac circuits$

we de"ine phasors & vectors whose length is the

ma(imum voltage or current$ and which rotate

around an origin with the angular speed o" the

oscillating current.The instantaneous

value o" the voltage or

current represented

%y the phasor is its

pro)ection on the y

a(is.

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The voltage and current in an ac circuit %oth

average to 'ero$ making the average useless indescri%ing their %ehavior.

We use instead the root mean s*uare +rms, we

s*uare the value$ "ind the mean value$ and then

take the s*uare root!

/0 volts is the rms value o" household ac.

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1y calculating the power and "inding theaverage$ we see that!

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Electrical "ires can %e started %y improper or

damaged wiring %ecause o" the heat caused %y a

too-large current or resistance.

A "use is designed to %e the hottest point in the

circuit & i" the current is too high$ the "use melts.

A circuit %reaker is similar$ e(cept that it is a

%imetallic strip that %ends enough to %reak the

connection when it %ecomes too hot. When itcools$ it can %e reset.

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A ground "ault circuit interrupter can cut o"" thecurrent in a short circuit within a millisecond.

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ow is the rms current in the capacitor

related to its capacitance and to the

"re*uency2 The answer$ which re*uirescalculus to derive!

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In analogy with resistance$ we write!

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The voltage andcurrent in a capacitor

are not in phase. The

voltage lags %y 304.

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RCCircuits

In an RCcircuit$ the current across the resistor

and the current across the capacitor are not in

phase. This means that the ma(imum current isnot the sum o" the ma(imum resistor current

and the ma(imum capacitor current they do

not peak at the same time.

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RCCircuits

This phasor diagram

illustrates the phaserelationships. The

voltages across the

capacitor and across the

resistor are at 304 in thediagram i" they are

added as vectors$ we

"ind the ma(imum.

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RCCircuits

This has the e(act same "orm as V5 IRi" wede"ine the impedance$ Z!

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RCCircuits

There is a phase angle%etween the voltage and

the current$ as seen in the

diagram.

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RCCircuits

The power in the circuit is given %y!

1ecause o" this$ the "actor cos is calledthe power "actor.

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6ust as with capacitance$ we can de"ine

inductive reactance!

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The voltage across an inductor leads the

current %y 304.

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The power "actor "or an RLcircuit is!

Currents in resistors$

capacitors$ and

inductors as a"unction o"

"re*uency!

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RLCCircuits

A phasor diagram is a use"ul way to analy'e an

RLCcircuit.

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RLCCircuits

The phase angle "or an RLCcircuit is!

I" XL=XC$ the phase angle is 'ero$ and the

voltage and current are in phase.

The power "actor!

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RLCCircuits

At high "re*uencies$ the capacitive reactance is

very small$ while the inductive reactance is verylarge. The opposite is true at low "re*uencies.

R i El t i l Ci it

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Resonance in Electrical CircuitsI" a charged capacitor is connected across an

inductor$ the system will oscillate inde"initely in

the a%sence o" resistance.

R i El t i l Ci it

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The rms voltages across the capacitor and

inductor must %e the same there"ore$ we cancalculate the resonant "re*uency.

R i El t i l Ci it

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In an RLCcircuit with an ac power source$ the

impedance is a minimum at the resonant"re*uency!

R i El t i l Ci it

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The smaller the resistance$ the larger the

resonant current!

#

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#ummary

The voltage "rom an ac generator varies

sinusoidally!

7hasor represents voltage or current in ac

circuit as it rotates$ its ycomponent gives theinstantaneous value.

Root mean s*uare +rms, o" a sinusoidally

varying *uantity!

#ummary

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#ummary

rms current in a capacitor!

Capacitive reactance!

Voltage across capacitor lags current %y 304

Impedance in an RCcircuit!

Average power!

#ummary

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#ummary

Inductive reactance!

Impedance o" an RLcircuit!

Impedance o" an RLCcircuit!

Resonant "re*uency o" an LCcircuit!

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